Almost all the Science Pupil are well aware of the term Integration. It is a common term in Mathematics. It also plays a very important role in the theory of Electromagnetics. In this article, I am going to discuss the formal meaning of integration and specifically what is Line Integration and its significance in the Electromagnetics.
In Mathematics, formally the Integration gives the area under the curve. Consider the integration shown in the figure below.
The differential or infinitesimal length is “dx” and corresponding to this small length an infinitesimal rectangular strip is shown. The area of this strip can be considered as multiplication of the value of f(x) at that x and length dx. In the process of integration, we add all such infinitesimal strips from A to B to give the area under the curve.
In Electromagnetics, we deal with various vector fields like Electric field, Magnetic field etc. Consider any such vector field E as shown in the following figure. Say, a line or vector AB is present within that field.
If we want the product of the length of the vector and component of the field along this vector (line) at every point of line then we write the integration as follows: –
This integration is termed as a line integration.
The significance of Line Integration in Electromagnetics
The formula for work in Electromagnetics is based on the concept of line integration. Say we are moving a charge in the electric field, then work done is calculated by
Basically, the line integration is required when we want to calculate the component or value of any vector field along the given vector path as in Ampere’s law.
Different types of Differential lengths
In Electromagnetics we deal with Cartesian Coordinate System, Cylindrical Coordinate System, and Spherical Coordinate System. So the infinitesimal length “dl” should be defined in each of the coordinate systems. As we deal with vectors and vector fields in Electromagnetics, “dl” consists of infinitesimal change along each axis of the respective coordinate system.
Suggested Reads: –